Non-dissipative Entropy Satisfying Discontinuous Reconstruction Schemes for Hyperbolic Conservation Laws

In this paper, we derive non-dissipative stable and entropy satisfying finite volume schemes for scalar PDEs. It is based on the previous analysis of [20], which deals with general reconstruction schemes. More precisely, we develop discontinuous-in-cell reconstruction schemes, based on a discontinuous reconstruction of the solution in each cell of the mesh at each time step. The intend is to handle well with discontinuous solutions. The schemes satisfy L-stability, decrease of the total variation and of an entropy, and thus are convergent to an entropy solution. A link with other formalisms is established. We propose a detailed numerical study in the cases of advection with constant velocity, of Burgers’ equations, and finally for a non-convex flux.